We derive explicit formulae to quantify the Markov chain state-space compression, or lumping, that can be achieved in a broad range of dynamical processes on real-world networks, including models of epidemics and voting behaviour, by exploiting redundancies due to symmetries. These formulae are applied in a large-scale study of such symmetry-induced lumping in real-world networks, from which we identify specific networks for which lumping enables exact analysis that could not have been done on the full state-space. For most networks, lumping gives a state-space compression ratio of up to ${10}^7$, but the largest compression ratio identified is nearly ${10}^{12}$. Many of the highest compression ratios occur in animal social networks. We also present examples of types of symmetry found in real-world networks that have not been previously reported.

我们推导出明确的公式来量化马尔可夫链状态空间压缩或集总，这可以通过利用对称性造成的冗余，在现实世界网络的广泛动态过程中实现，包括流行病和投票行为模型。这些公式应用于对现实世界网络中这种对称性诱导集总的大规模研究，从中我们确定了特定网络，集总可以对其进行精确分析，而这在完整的状态空间上是无法完成的。对于大多数网络，集总给出的状态空间压缩比高达${10}^7$，但确定的最大压缩比几乎是 ${10}^{12}$. 许多最高压缩率出现在动物社交网络中。我们还展示了在现实世界网络中发现的以前没有报道过的对称类型的例子。